Fogging problem (i.e.) screen is foggy at start up but clears 20 to 30 mins use.

The Sony TV is NOT completely defective. The image projected is just not clear. We have had this problem before and our resolution was to wait for the image to clear up. This took about 20-30 minutes time and all are normal.

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IF I understand the problem from the limited info in your question, the problem is a distorted picture on whatever surface [probably a wall, or screen] onto which the image is projected.

With all lenses, the axis [centerline of the lens projected] MUST be absolutely PERPENDICULAR [in both the vertical and horizontal planes] to the perfectly plane [flat] surface onto which the image is projected.IF for example, the projector is mounted in a rear corner of the room, and the camera is aimed downward and to one side to project the image on the center of the opposite wall, or a screen located there, the image will be distorted. It will be larger on one edge [fartherist from the lens] and smaller on the opposite [nearest] edge.If it is non-perpendicular in only one axis, then two opposite edges will be different lengths, and if out of perpendicular in two axis then no edges will be the same length.The solution to this problem is that the center axis of the lens must be perpendicular to the surface projected on, in both vertical and horizontal axis.

Are you Technician ?
You should call Samsung. The problem is in the Digital printer circuit board where is the DVI input connector. It have a few of regulator 3,3V and surface mount ellectrolytics capacitors were leaked and when the unit get hot the regulator get lower then 3V that moment you get the pixel block on your screen
Depend your country and depend Technician if they can repair at component level or PCB level that mean they replace a complet board. In Australia it cost between $350 to $800. I hope that it give you some ideas
Cheers

Something I found interesting from the internet - lamp life isn't
when the bulb dies, but when a bulb has lost 10% of its brightness - so
you might well be able to reset the timer and get much more use from
the bulb - the image just wont be as bright.

The lamp may not have been changed - just the timer reset - How many hours has the lamp really done - 1000, 3000, 5000?If you know that the hours are genuine, then was the quartz bulb contaminated with a touch of oil from skin contact?This will substantially shorten lamp life, due to the quartz going milky and opaque.The easiest way to check, is to pull the lamp and have a look at the bulb.

If need be replace the lamp and reset the lamp timer.

These instructions are for a Sony Cineza VPL HS3, but may be the same for the HS51
1st thing to check - is the cover properly/completely closed - this can stop the projector from working.
Connect the power and set the projector to standby mode.
Press these buttons on the remote control in this order, for less than 5 seconds each.
Reset - Left Arrow - Right Arrow - Enter

You probably just need a new lamp. If your tv is that old, im sure you do. I just got a new one from Projector Lamp Services (relampit.com) it works great... Only took me a few mins to install it too. I missed my TV!!!

if the lines on your screen looks anything like the image below, you do have a problem with the lcd, problem is you need to replace what is called the light engin. unfortunatly the price of this part is just about as much as a new replacement tv. Get a sceond opinion fron a sony service center near you, as i cannot see your picute. good luck